Nineteenth Century Philosophy

Philosophy of Mathematics and Logic

Did Gottlob Frege succeed in reducing mathematics to logic?

Alas, no. When the second volume to Frege’s Basic Laws of Arithmetic (1893) had been sent to the printer, he received a letter from British philosopher, historian, and mathematician Bertrand Russell (1872–1970) in which Russell introduced his famous paradox: “Is the class of all classes that are not members of itself a member of itself or not?” The question is coherent but it entails a contradiction, so it has no answer.

Frege had to admit that he had no foundation for his reasoning: “A scientist can hardly encounter anything more undesirable than to have the foundation collapse just as the work is finished. I was put in this position by a letter from Mr. Bertrand Russell when the work was almost through the press.” The great irony in this is that Russell embarked on his own project to reduce mathematics to logic—and failed!